NUMBER FIELD ZETA INTEGRALS and L-FUNCTIONS the Basic Zeta

NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

The basic zeta function is X ζ(s) = n−s, Re(s) > 1. n≥1 Riemann proved that the completed basic zeta function, Z(s) = π−s/2Γ(s/2)ζ(s), Re(s) > 1, has an entire meromorphic continuation that satisﬁes the functional equation Z(1 − s) = Z(s), s ∈ C. One of his proofs proceeds by arguing as follows. • Poisson summation shows that the theta function, X 2 θ(t) = e−πin t, t > 0, n∈Z is a modular form, θ(1/t) = t1/2θ(t). • Z is the Mellin transform (essentially a Fourier transform) of θ, Z 1 s/2 dt Z(s) = 2 (θ(t) − 1)t , Re(s) > 1. t>0 t As t → ∞ the integrand decays rapidly regardless of the value of s, posing no obstacle to the integral’s convergence at its improper upper limit of integration. But as t → 0+ the integrand is O(t(Re(s)−1)/2 dt/t), requiring Re(s) > 1 for the integral to converge at its improper lower limit. • The transformation law for θ shows that in fact Z 1 s/2 (1−s)/2 dt 1 1 Z(s) = 2 (θ(t) − 1)(t + t ) − − . t>1 t s 1 − s With the problematic lower limit of integration no longer present, the right side is now sensible as a meromorphic function for all s ∈ C, and it visibly satisﬁes the functional equation. Similar methods apply to any Dirichlet L-function, X L(s, χ) = χ(n)n−s, Re(s) > 1. n≥1 The Dirichlet L-function has a suitable completion Λ(s, χ), Re(s) > 1, involving a factor similar to the factor π−s/2Γ(s/2) for the basic zeta function, but now taking into account the parity and the conductor of χ. Again the completion has a continuation and a functional equation. The calculation now involves a Gauss sum along with the other ingredients, and the functional equation includes a factor called a root number, W (χ)Λ(1 − s, χ) = Λ(s, χ). 1 2 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

If the Dirichlet character is quadratic then the root number W (χ) is 1. Both the basic zeta function and the Dirichlet L-function are defined over the basic number field Q. The technique of completion, continuation, and functional equation apply more generally to the Dedekind zeta function of a number field k,

X −s ζk(s) = Na , a and even for any L-function associated to a number field k and a so-called Hecke character of k, X −s Lk(s, χ) = χ(a)Na . a But the definition of a Hecke character is complicated, and the calculations and the root number become ever more elaborate as the environment grows. As Artin, Iwasawa, Tate, and perhaps others knew around 1950, working in the environment of the adeles simplifies the calculation decisively. In the adelic environment, the definition of a Hecke character is simple and natural, and the root number emerges naturally from local calculations. After defining the adelic zeta integral and looking at some of its local factors, these notes establish a global adelic version of its continuation and functional equation. Strikingly, the procedure is essentially identical to Riemann’s original argument. The exposition of this material is based on a presentation and written materials by Paul Garrett. Then the notes continue to discuss the local factors of the zeta integral, following Kudla’s lecture in the Introduction to the Langlands Program volume.

Contents

Part 1. The Zeta Integral 2 1. Deﬁnition of the Zeta Integral 2 2. Local Zeta Integrals 3 2.1. Unramiﬁed nonarchimedean places 3 2.2. Real archimedean places 4 2.3. Complex archimedean places 4

Part 2. Global Theory 5 3. Convergence 5 4. Elements of Adelic Harmonic Analysis 6 5. Continuation and Functional Equation 8

Part 3. Local Theory 10 6. The Local Zeta Integral 10 7. Spaces of Distributions 11 7.1. The nonarchimedean unramiﬁed case 13 7.2. The nonarchimedean ramiﬁed case 14 7.3. The nonarchimedean trivial case 15 8. The Local Functional Equation 16 8.1. Additive characters 16 8.2. The local functional equation 18 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 3

8.3. Pre-normalizing the ε-factor calculation 18 8.4. The ε-factor calculation 19 9. The Global ε-Factor 21

Part 4. The Final Calculation 22

Part 5. Application: Quadratic ε-Factors are Trivial 22

Part 1. The Zeta Integral 1. Definition of the Zeta Integral Let k be a number ﬁeld, and let A be its adele ring. Recall that k is discrete in A and the quotient A/k is compact. Let J be the idele group of k. Recall that k× is discrete in the unit idele group J1 and the quotient J1/k× is compact. (The statement here is Fujisaki’s Lemma, encompassing both the structure theorem for the units group of the integers of k and ﬁniteness of the class number of k. The adelic repackaging of those results as Fujisaki’s Lemma is optimal for our purposes here.) Let s be a complex parameter, let × χ : J −→ C be a Hecke character of J, and let ϕ : A −→ C be an adelic Schwartz function. Let | | denote the idele norm, and let d× denote Haar measure on J. Then the adelic zeta integral associated to χ and ϕ is Z Z(s, χ, ϕ) = |α|sχ(α)ϕ(α) d×α. J Here s and χ combine to encode the overall character χ | · |s being integrated against the Schwartz function. Since χ need not be discretely parameterized, the parame- ters of the zeta integral thus incorporate some redundancy: Z(s0 + s, χ, ϕ) = Z(s0, χ | · |s, ϕ) for all s0, s, χ. The integral converges for values of s in a right half-plane.

2. Local Zeta Integrals The Hecke character in the zeta integral decomposes as a product of local characters, O × χ = χv : J −→ C . v The assertion that the Schwartz function in the zeta integral decomposes similarly, O ϕ = ϕv, each ϕv is Schwartz, v is not quite true, since in fact ϕ is a ﬁnite sum of such products, but we treat ϕ as a monomial from now on without further comment. For a nonarchimedean place v, 4 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS the local Schwartz space consists of the compactly supported locally constant functions on kv, and for an archimedean place v the local Schwartz space consists of the smooth functions on kv all of whose derivatives are rapidly decreasing. In consequence of χ and ϕ factoring, the global zeta integral Z Z(s, χ, ϕ) = |α|sχ(α)ϕ(α) d×α J immediately takes the form of an Euler product of local zeta integrals, Y Z(s, χ, ϕ) = Zv(s, χv, ϕv) v where for each place v, Z s × Zv(s, χv, ϕv) = |α|vχv(α)ϕv(α) dv α. × kv The Euler factorization will play no role in the global continuation and functional equation argument. However, we now examine some local Euler factors.

2.1. Unramified nonarchimedean places. Let v be a finite place of k, and let Ov be the ring of integers of kv. With v fixed, freely drop it from the notation. Let the local character have no discretely-parameterized part,

× × s χ : k −→ C , χ(α) = |α| o (for any global Hecke character, this holds at almost all the ﬁnite places), and let the local Schwartz function be the characteristic function of the local integers,

ϕ : k −→ C, ϕ(α) = 1O(α). To compute the local integral Z Z(s) = |α|sχ(α)ϕ(α) d×α, k× take a uniformizer (valuation-1 element) $. Thus |$| = Np−1 where p is the prime ideal of Ok corresponding to the place v. Normalize the multiplicative Haar measure so that O× has measure 1. Then the local integral is, allowing a slight abuse of notation at the last step,

∞ Z Z X Z(s) = |αη|s+so ϕ(αη) d×η d×α = |$`|s+so = (1 − χ(p)Np−s)−1. × × × k /O O `=0 Thus the local zeta integral gives an Euler factor of the Hecke L-function at almost all places. And the local Schwartz function is its own Fourier transform, so that at such places there is no need to replace it by its Fourier transform in the local factor of the functional equation. The case of a ramiﬁed local character in the nonarchimedean case will be discussed later in the writeup. NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 5

2.2. Real archimedean places. Let kv = R. Consider the trivial character and the Gaussian Schwartz function × × χ : R −→ C , χ(α) = 1, −πα2 ϕ : R −→ C, ϕ(α) = e . The local integral is Z Z ∞ s × s −πt2 dt − s s Z(s) = |α| ϕ(α) d α = 2 t e = π 2 Γ . × t 2 R 0 This is the familiar archimedean factor from completing the basic zeta function or an even Dirichlet L-function. On the other hand, consider the sign character (which arises as the inﬁnite part of an odd Dirichlet character viewed as a classical Hecke character, for example) and the simplest odd Schwartz function that incorporates the Gaussian, × × χ : R −→ C , χ(α) = sgn(α), −πα2 ϕ : R −→ C, ϕ(α) = αe . Then the local integral is Z Z ∞ s × s+1 −πt2 dt − s+1 s + 1 Z(s) = |α| χ(α)ϕ(α) d α = 2 t e = π 2 Γ . × t 2 R 0 This is the archimedean factor from completing an odd Dirichlet L-function.

2.3. Complex archimedean places. Let kv = C. The unitary characters are α m χ : × −→ , χ(α) = where m ∈ . C T |α| Z The Haar measure is d+α r dr dθ d×α = = . |α|2 r2 Note that the change of variable ρ = r2 gives 2r dr/r2 = dρ/ρ. 2 If m = 0 then let ϕ(α) = e−π|α| . The local integral is Z Z ∞ s × 2s −πr2 r dr −s Z(s) = |α| χ(α)ϕ(α) d α = 2π r e 2 = π · π Γ(s). × C r C 0 2 If m > 0 then let ϕ(α) =α ¯me−π|α| . The local integral is Z Z ∞ s × 2s+m −πr2 r dr Z(s) = |α| χ(α)ϕ(α) d α = 2π r e 2 × C r C 0 −s− m m = π · π 2 Γ s + . 2 2 If m < 0 then let ϕ(α) = α−me−π|α| . The local integral is Z Z ∞ s × 2s−m −πr2 r dr Z(s) = |α| χ(α)ϕ(α) d α = 2π r e 2 × C r C 0 −s+ m m = π · π 2 Γ s − . 2 Thus in all cases, −s− |m| |m| Z(s) = π · π 2 Γ s + . 2 6 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

A more classical normalization is to multiply the measure by 2/π and to replace 2 2 e−π|α| by e−2π|α| in the Schwartz functions. Then in particular when m = 0, the local integral is 2 · (2π)−sΓ(s), and by Legendre’s duplication formula Γ(z)Γ(z + 1/2) = 21−2zπ1/2Γ(2z) this is the product of the even and odd real archimedean factors from a moment ago. The classical normalization ﬁts tidily with the factorization of the Dedekind zeta function of a CM-extension.

Part 2. Global Theory This part of the writeup establishes the global continuation and functional equation for the adelic zeta integral. The method is laid out to look similar to Riemann’s original argument.

3. Convergence This section shows that the half-zeta integral Z |α|sχ(α)ϕ(α) d×α |α|≥1 converges for all s ∈ C. We may take the Schwartz function to be monomial, N ϕ = v ϕv, since any Schwartz function is a ﬁnite sum of such. The Schwartz function decays rapidly, Y −n + |ϕ(α)| ≤ Cϕ,n sup{|αv|v, 1} , α ∈ A, n ∈ Z . v (The product is nontrivial at only ﬁnitely many terms, and similarly for products to come in this section except the last one.) Since sup{r, 1} = r1/2 sup{r1/2, r−1/2} for any r > 0, the bounds become (omitting constants from now on) −n/2 Y 1/2 −1/2 −n |ϕ(α)| . |α| sup{|αv|v , |αv|v } v −n/2 Y n/2 −n/2 + = |α| inf{|αv|v , |αv|v }, α ∈ J, n ∈ Z , v and so furthermore Y n/2 −n/2 + |ϕ(α)| . inf{|αv|v , |αv|v }, |α| ≥ 1, n ∈ Z . v We may assume that χ is discretely-parameterized by absorbing its continuous parameter into s. Thus, letting σ = Re(s),

Z Z s × σ Y n/2 −n/2 |α| χ(α)ϕ(α) d α . |α| inf{|αv|v , |αv|v } |α|≥1 |α|≥1 v Z Y σ n/2 −n/2 × + ≤ |αv| inf{|αv|v , |αv|v } d α, n ∈ Z . × v kv Each nonarchimedean integral is a sum over Z that converges absolutely in both directions for n > 2|σ|. Speciﬁcally, it is X X 1 − q−n q−e(σ+n/2) + qe(σ−n/2) = v . v v −σ−n/2 σ−n/2 e≥0 e>0 (1 − qv )(1 − qv ) NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 7

The archimedean integrals are similar. Thus for any s the product is dominated by the convergent product for ζk(σ + n/2)ζk(−σ + n/2)/ζk(n) for all suﬃciently large n.

4. Elements of Adelic Harmonic Analysis Let T denote the circle group of complex numbers of absolute value 1. Take an adelic additive unitary character ψ : A −→ T that is trivial on k, ψ(k) = 1. The adelic Fourier transform of a Schwartz function ϕ : A −→ C is Z Fϕ : A −→ C, (Fϕ)(x) = ϕ(y)ψ(−xy) dy. A Here the Haar measure on A is normalized so that (FFϕ)(x) = ϕ(−x). We construct such a character, if only to show that one exists and perhaps to give the reader some grounding. First define an additive unitary character for each rational prime. For a finite prime p of Q the character is ep(x) = exp(−2πix), x ∈ Qp, whose meaning and continuity are clear on the dense subset Z[1/p] of Qp and therefore on all of Qp. For the infinite prime of Q the character is e∞(x) = exp(2πix), x ∈ R. N Thus p ep(Q) = 1, the product including the infinite prime. Now return to the general number field k, and take each local character ψv to be the trace followed by the corresponding character, ψ = e ◦ tr , v | p. v p kv /Qp

(Thus the kernel of ψv is the local inverse different, simply the local integers away from ramification.) The product of the local characters, O ψ = ψv : A −→ T, v is an adelic additive unitary character. Not only does the general adelic additive unitary character decompose as a product of local characters in this fashion, but in fact we will later see that the general adelic additive unitary character can be normalized to this particular ψ. For any Schwartz function ϕ and any idele α, define the dilation

ϕα : A −→ C, ϕα(x) = ϕ(αx). The relation between the Fourier transforms of the dilation and of the original function is −1 F(ϕα) = |α| (Fϕ) −1 . A α To see this, compute Z Z −1 (F(ϕα))(x) = ϕα(y)ψ(−xy) dy = ϕ(αy)ψ(−α xαy) d(αy)/|α|A A A = |α|−1(Fϕ)(α−1x). A 8 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

The adelic Poisson summation formula is derived as follows. Given a Schwartz function f : A −→ C, symmetrize to obtain a k-periodic function X F : A −→ C,F (x) = f(x + k). k∈k The symmetrized function is equal to its Fourier series, X F (x) = ck(F )ψ(kx), k∈k where the kth Fourier coeﬃcient is Z ck(F ) = F (y)ψ(−ky) dy A/k Z X = f(y + κ)ψ(−k(y + κ)) dy A/k κ∈k Z = f(y)ψ(−ky) dy since ψ is additive and ψ(k) = 1 A = (Ff)(k). This gives the Poisson summation formula, X X f(x + k) = (Ff)(k)ψ(kx), k∈k k∈k and especially when x = 0 we have X X f(k) = (Ff)(k). k∈k k∈k

Let A be a topological group (locally compact and countably-based) and B a closed subgroup. The mock-Fubini formula is Z Z Z f(α) dα = F (α) dα where F (α) = f(αβ) dβ. A A/B B

5. Continuation and Functional Equation Again, for a Hecke character χ : J −→ C× and a Schwartz function ϕ : A −→ C, the associated adelic zeta integral is Z Z(s, χ, ϕ) = |α|sχ(α)ϕ(α) d×α, J convergent for s in a right half-plane. To establish its analytic continuation and functional equation, begin by deﬁning an adelic theta function, X θϕ(α) = ϕ(αk), α ∈ J. k∈k The adelic Poisson summation formula and then the formula for the Fourier transform of a dilation give the transformation law for the theta function, X X −1 X −1 −1 θϕ(α) = ϕα(k) = (F(ϕα))(k) = |α| (Fϕ)α−1 (k) = |α| θFϕ α . k∈k k∈k k∈k NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 9

A calculation that uses the mock-Fubini formula at the ﬁrst step shows that the theta function contributes to the zeta integral, Z X Z(s, χ, ϕ) = |αk|sχ(αk)ϕ(αk) d×α × J/k k∈k× ! Z X = |α|sχ(α) ϕ(αk) − ϕ(0) d×α × J/k k∈k Z s × = |α| χ(α) θϕ(α) − ϕ(0) d α. × J/k The zeta integral splits into two terms. Let × × S = {α ∈ J : |α| ≤ 1}/k ,T = {α ∈ J : |α| ≥ 1}/k . Then S ∪ T = J/k× and S ∩ T has measure 0, so that Z s × Z(s, χ, ϕ) = |α| χ(α) θϕ(α) − ϕ(0) d α S Z s × + |α| χ(α) θϕ(α) − ϕ(0) d α. T The transformation law for the theta function shows that the ﬁrst term of the right side of the previous display is Z s × |α| χ(α) θϕ(α) − ϕ(0) d α S Z s−1 −1 × = |α| χ(α) θFϕ(α ) − (Fϕ)(0) d α S (1) Z − ϕ(0) |α|sχ(α) d×α S Z + (Fϕ)(0) |α|s−1χ(α) d×α. S And substituting α−1 for α shows that the main term of the right side of (1) is Z s−1 −1 × |α| χ(α) θFϕ(α ) − (Fϕ)(0) d α S Z 1−s −1 × = |α| χ (α) θFϕ(α) − (Fϕ)(0) d α. T To study the other two terms of the right side of (1), again use the mock-Fubini formula Z Z Z f(α) dα = F (α) dα where F (α) = f(αβ) dβ. A A/B B Let A = S, B = J1/k×, and f(α) = |α|sχ(α). Then the inner integral is Z Z F (α) = |αβ|sχ(αβ) d×β = |α|sχ(α) χ(β) d×β 1 × 1 × J /k J /k ( |α|svol( 1/k×) if χ = 1, = J 0 if χ 6= 1. 10 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

Here we are using the fact that B is a compact group, although A isn’t a group at all. And since A/B =∼ (0, 1], where the map is α 7→ |α|, the nontrivial case of the outer integral is (omitting constants for the moment) Z Z 1 dt 1 |α|s d×α = ts = . 1 × t s S/(J /k ) 0 Thus the second term of the right side of (1) is  ϕ(0) Z −vol( 1/k×) if χ = 1, −ϕ(0) |α|sχ(α) d×α = J s S 0 if χ 6= 1. Similarly the third term of the right side of (1) is  (Fϕ)(0) Z −vol( 1/k×) if χ = 1, (Fϕ)(0) |α|s−1χ(α) d×α = J 1 − s S 0 if χ 6= 1. This analysis gives the desired properties of the zeta integral. If χ = 1 then Z s 1−s × Z(s, 1, ϕ) = |α| θϕ(α) − ϕ(0) + |α| θFϕ(α) − (Fϕ)(0) d α T ϕ(0) (Fϕ)(0) − vol( 1/k×) + , J s 1 − s while if χ 6= 1 then the two terms that contribute poles vanish, and so Z s 1−s −1 × Z(s, χ, ϕ) = |α| χ(α) θϕ(α) − ϕ(0) + |α| χ (α) θFϕ(α) − (Fϕ)(0) d α. T In both cases, the integral is an entire function of s, as discussed earlier. In ei- ther case, the zeta integral Z(s, χ, ϕ) is visibly invariant under the substitution (s, χ, ϕ) 7→ (1 − s, χ−1, Fϕ). When χ = 1, it is meromorphic with simple poles at s = 0 and s = 1, and when χ 6= 1, it is entire. That is, we have established that • Z(s, χ, ϕ) has a continuation to the complex plane, and the continuation satisﬁes the functional equation Z(s, χ, ϕ) = Z(1 − s, χ−1, Fϕ). • The continuation of Z(s, χ, ϕ) is analytic when χ 6= 1, and it is meromorphic with simple poles at s = 0 and s = 1 when χ = 1. Later in this writeup we will view the zeta integral as an (s, χ)-parameterized function Z(s, χ) of Schwartz functions ϕ, i.e., as a distribution. From this point of view, the functional equation is Z(s, χ) = FZ(1 − s, χ−1). This is the version of the functional equation that will be quoted at the very end of the writeup.

Part 3. Local Theory Each local factor of the zeta integral can be continued as well. Doing so leads naturally to an L-function and an ε-factor associated to the Hecke character χ. Eventually, combining the local results with the global work produces a continuation and a function equation for the L-function. NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 11

6. The Local Zeta Integral We work over a local ﬁeld F . Thus F is nonarchimedean, or F = R, or F = C. Let S(F ) denote the space of Schwartz functions on F . With F clear in our minds we freely suppress it from the notation, for example writing S rather than S(F ). Let × × χ : F −→ C be a character, not necessarily unitary. For any Schwartz function ϕ ∈ S and for a complex parameter s ∈ C, the local zeta integral is formally (not yet discussing convergence) Z Z(s, χ, ϕ) = ϕ(x)χ(x)|x|s d×x. F × In the archimedean case, the multiplicative Haar measure in the deﬁnition is ( √ dx is Lebesgue measure and |x| = x2 if F = , d×x = dx/|x| where R dx is twice Lebesgue measure and |x| = xx if F = C. In the nonarchimedean case the multiplicative Haar measure is determined up to constant multiple, and soon we will normalize it.

7. Spaces of Distributions Introduce two spaces: • The Schwartz functions that vanish to all orders at 0, (n) S0 = {ϕ ∈ S : ϕ (0) = 0 for all n ≥ 0}. • The Taylor expansions of Schwartz functions at 0, X T = { (ϕ(n)(0)/n!)xn : ϕ ∈ S}. n≥0

(If F is nonarchimedean then S0 is simply the Schwartz functions that vanish at 0, and T = {ϕ(0) : ϕ ∈ S} is simply a copy of C.) Then we have a short exact sequence inc 0 −→ S0 −→ S −→ T −→ 0. Introduce two more spaces: • The tempered distributions that are supported at 0, 0 0 S{0} = {u ∈ S : supp(u) ⊂ {0}}. These are the ﬁnite linear combinations δ-derivatives, counting δ as its own 0 0 zeroth derivative. Note that S{0} = T . • The restrictions of tempered distributions to S0, S0 = {u : u ∈ S0}. S0 S0 By the Hahn–Banach Theorem, S0 is all of S0 . S0 0 The previous short exact sequence dualizes to another, 0 −→ S0 −→ S0 −→res S0 −→ 0. {0} S0 × For any a ∈ F , the right-translation operator Ra on Schwartz functions ϕ ∈ S is deﬁned by the condition (Raϕ)(x) = ϕ(xa). 12 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

(Since F is a ﬁeld, right-translation and left-translation are the same, but the rule Rab = Ra ◦ Rb holds with no reference to commutativity, whereas the general rule for left-translation is Lab = Lb◦La.) The right-translation operator Ra on tempered distributions u ∈ S0 is deﬁned as an adjoint,

hRau , ϕi = hu , Ra−1 ϕi.

(The inverse ensures that again the rule Rab = Ra ◦ Rb holds with no reference to commutativity.) A tempered distribution u ∈ S0 is an eigendistribution if for some character χ : F × −→ C×, not necessarily unitary, × Rau = χ(a)u for all a ∈ F . The eigendistributions having one particular character χ form the eigenspace S0(χ). Let χ : F × −→ C× be a character, not necessarily unitary. The previous short exact sequence of distributions restricts to eigenspaces, 0 −→ S0 (χ) −→ S0(χ) −→res S0 (χ), {0} S0 where now the restriction map need not surject: a χ-eigendistribution on S0 lifts to a distribution on S, but the lift need not again be a χ-eigendistribution. The first object of the previous sequence is at most one-dimensional. Specifically (letting na stand for nonarchimedean),  0 Cδ = S{0}(χ0) if F = na,   M k M 0 −k  CD δ = S{0}(χk) where χk(x) = x if F = R, 0  S{0} = k≥0 k≥0   M k,` M 0 −k −`  CD δ = S{0}(χk,`) where χk,`(x) = x x if F = C.  k,`≥0 k,`≥0 We quote that the third object of the previous sequence is exactly one-dimensional. × Specifically, if we let λχ denote the distribution that integrates against χ over F , Z × λχ : ϕ 7−→ ϕ(x)χ(x) d x, F × then 0 S (χ) = λχ. S0 C Thus the sequence takes various possible forms: 0 F = na, χ 6= χo : 0 −→ 0 −→ S (χ) −→ Cλχ, 0 F = R, χ 6= χk : 0 −→ 0 −→ S (χ) −→ Cλχ, 0 F = C, χ 6= χk,` : 0 −→ 0 −→ S (χ) −→ Cλχ, F = na, χ = χ : 0 −→ δ −→ S0(χ ) −→ λ , o C o C λχo k 0 F = R, χ = χk : 0 −→ CD δ −→ S (χk) −→ Cλχk , k,` 0 F = C, χ = χk,` : 0 −→ CD δ −→ S (χk,`) −→ Cλχk,` . In the first three cases dim(S0(χ)) ≤ 1, and in the last three dim(S0(χ)) ≤ 2. The result that we want is: In all cases, dim(S0(χ)) = 1. That is: NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 13

• In the ﬁrst three cases, where S0(χ) injects into S0 (χ), we need to show S0 0 0 0 that it surjects by showing that a lift of λχ from S (χ) to S lies in S (χ). S0 • In the last three cases, where the map from S0(χ) to S0 (χ) has a one- S0 dimensional kernel, we need to show that its image is trivial by showing 0 0 0 that no lift of λχ from S (χ) to S lies in S (χ). S0 In sections 7.1–7.3 we will establish the result in the nonarchimedean case modulo the invocation that the right-hand link of the sequence is one-dimensional, meeting some fruitful ideas in the process. Then, introducing Fourier analysis in section 8.1 will give two nonzero basis elements of S0(χ) in section 8.2, so that the two must be proportional. The constant of proportionality is the local ε-factor, and we will compute it in sections 8.3–8.4. 7.1. The nonarchimedean unramiﬁed case. Now assume that F is nonarchimedean. Let O = the ring of integers of F, P = the maximal ideal of O, $ = a generator of P, q = |O/P|, e −e × |u$ | = q for u ∈ O and e ∈ Z, Z d×x = multiplicative Haar measure, normalized so that d×x = 1. O× Consider a complex parameter s ∈ C and a character χ : F × −→ C×. As explained above, the distribution

hλ , ϕ0i = Z(s, χ, ϕ0) spans the one-dimensional eigenspace S0 (χ | · |s). Since each Schwartz function S0 ϕ0 ∈ S0 vanishes to all orders at 0, the integral is an entire function of s. However, now consider instead a Schwartz function ϕ from the larger space S rather than from S0. The local zeta integral remains analytic in s for Re(s) > 0, but for general s we can not immediately take the alleged distribution “ hλ , ϕi = Z(s, χ, ϕ) 00 as a basis element of S0(χ | · |s), or even as an element of S0 at all. To work around the problem, note that by the nature of nonarchimedean Schwartz functions, ϕ is constant on some neighborhood Pe−1 of 0. Thus the related function −1 ϕ0 = (1 − R$−1 )ϕ, ϕ0(x) = ϕ(x) − ϕ(x$ ) e is identically 0 on P , i.e., ϕ0 ∈ S0. Consequently, the local zeta integral of the related function, Z s × Z(s, χ, ϕ0) = ϕ0(x)χ(x)|x| d x = hλ , ϕ0i F × is entire in s. The distribution µ whose output is this related local zeta integral,

hµ , ϕi = hλ , ϕ0i = Z(s, χ, ϕ0), lies in S0(χ | · |s), because the preliminary calculation that for a ∈ F × and ϕ ∈ S,

(Ra−1 ϕ)0 = (1 − R$−1 )(Ra−1 ϕ) = Ra−1 (1 − R$−1 )ϕ = Ra−1 (ϕ0), 14 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS justiﬁes the third equality in the calculation

hRaµ , ϕi = hµ , Ra−1 ϕi = hλ , (Ra−1 ϕ)0i = hλ , Ra−1 (ϕ0)i s s = hRaλ , ϕ0i = χ(a)|a| hλ , ϕ0i = χ(a)|a| hµ , ϕi. So we have proved that dim(S0(χ | · |s)) ≥ 1, provided that µ is not the zero dis- o tribution. To show that indeed µ is not the zero distribution, let ϕ = 1O be the characteristic function of O, and compute Z Z o −1 s × × hµ , ϕ i = (1O(x) − 1O(x$ )χ(x)|x| d x = χ(x) d x. F × O× We are assuming that χ is unramiﬁed, and so (recalling that the multiplicative Haar measure of O× has been normalized to 1) the result is 1. No claim is made that the basis element µ lifts the local zeta integral distribu- 0 s 0 s tion λ from S (χ | · | )) to S (χ | · | )). It does not, because in general ϕ0 6= ϕ S0 for ϕ ∈ S0. We will see that no such lift is possible for the trivial character, and so in that case µ must be a multiple of the δ distribution. To see that in fact µ is exactly δ, compute that since any given ϕ ∈ S is constant on some Pe−1 we have Z Z hµ , ϕi = (ϕ(x) − ϕ(x$−1)) d×x = (ϕ(x) − ϕ(x$−1)) d×x F × F ×−Pe Z Z Z = ϕ(x) d×x − ϕ(x$−1) d×x = ϕ($e−1x) d×x F ×−Pe F ×−Pe O× = ϕ(0).

That is, µ = δ in the trivial character case as claimed.

We will continue the local zeta integral meromorphically to all s ∈ C, obtaining a second basis element of S0(χ | · |s) away from the poles of the continuation. The idea is to break the related integral Z(s, χ, ϕ0) into two pieces and recover the original Z(x, χ, ϕ) times a factor. Compute that for Re(s) > 0, Z Z Z s × s × −1 s × ϕ0(x)χ(x)|x| d x = ϕ(x)χ(x)|x| d x − ϕ(x$ )χ(x)|x| d x F × F × F × = (1 − χ($)q−s)Z(s, χ, ϕ) = L(s, χ)−1Z(s, χ, ϕ).

In terms of distributions, slightly rearranging the previous display gives

hλχ |·|s , ϕi = L(s, χ)hµχ |·|s , ϕi, Re(s) > 0. However, the right side of the previous display is the product of • an L-factor that is meromorphic in s, • and a local zeta integral that is entire in s. Thus we use the right side to continue the left side meromorphically in s,

hλχ |·|s , ϕi = L(s, χ)hµχ |·|s , ϕi, s ∈ C. And so L(s, χ) is naturally a ratio of two basis-elements of a one-dimensional distribution space, away from poles. NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 15

7.2. The nonarchimedean ramified case. If χ is ramified then for any integers m and n with m ≥ n, m−1 Z X Z χ(x)|x|s d×x = χ($)eq−es χ(x) d×x = 0. n m × P −P e=n O It follows that for any ϕ ∈ S (which is constant on some neighborhood of 0), the integral Z ϕ(x)χ(x)|x|s d×x, n 0 F ×−Pn is independent of n once n is large enough. That is, the improper integral Z Z(s, χ, ϕ) = ϕ(x)χ(x)|x|s d×x F × o −1 converges. Also, if we specify ϕ = 1O× · χ then Z Z(s, χ, ϕo) = d×x = 1, O× Thus the distribution λ ∈ S0 , hλ , ϕi = Z(s, χ, ϕ) S0 is nonzero and immediately extends from S0 (χ | · |s) to S0(χ | · |s). The natural S0 definition of the local L-factor in the ramified case is therefore L(s, χ) = 1. We have now shown that dim(S0(χ)) ≥ 1 for all χ in the nonarchimedean case, unramified or ramified, and so dim(S0(χ)) = 1 for all nontrivial χ in the nonarchimedean case. 7.3. The nonarchimedean trivial case. In the nonarchimedean case, we still 0 need to prove that S (χo) is only one-dimensional. Let ϕ denote a generic Schwartz function. Recall the first short exact sequence from earlier, now augmented by a left inverse of its inclusion map, inc / 0 /S0 o S /T /0. 1−∗(0)1O

Here 1 − ∗(0)1O denotes the map from S to S0 that takes any ϕ to ϕ − ϕ(0)1O, and so (1 − ∗(0)1O) ◦ inc = 1 on S0. As before, dualize and quote the Hahn–Banach Theorem to get a second short exact sequence, now augmented by a right inverse of its restriction map, res / 0 / 0 / 0 / 0 S{0} S o S 0 0 ∗ S0 (1−∗(0)1O )

∗ 0 where res ◦ (1 − ∗(0)1O) = 1 on S . S0 0 Let λ = λχo ∈ S (χo) be the basic integration distribution on S0, S0 Z × hλ , ϕ0i = ϕ0(x) d x, ϕ0 ∈ S0, F × and let λ˜ ∈ S0 denote the corresponding lift from the sequence, ˜ ∗ 0 λ = (1 − ∗(0)1O) λ ∈ S . 16 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

That is to say, ˜ hλ , ϕi = hλ , ϕ − ϕ(0)1Oi, ϕ ∈ S. The general lift of λ to S0 is λ˜ + cδ where c ∈ C. We want to show that: No lift of λ to S0 is F ×-invariant. But δ is F ×-invariant, and so it suﬃces to show that the particular lift λ˜ is not F ×-invariant. Compute that for any a ∈ F × and any ϕ ∈ S0, ˜ ˜ hRaλ , ϕi = hλ , Ra−1 ϕi = hλ , Ra−1 ϕ − (Ra−1 ϕ)(0)1Oi = hλ , Ra−1 ϕ − ϕ(0)1Oi ˜ (showing by a small exercise that Raλ is again a lift of λ), and in particular for a = 1, ˜ hλ , ϕi = hλ , ϕ − ϕ(0)1O)i.

Thus (noting that if ϕ ∈ S then (1 − Ra−1 )ϕ ∈ S0) Z ˜ −1 × h(1 − Ra)λ , ϕi = hλ , (1 − Ra−1 )ϕi = ϕ(x) − ϕ(xa ) d x. F ×

The integral is in general nonzero. For example, choose ϕ = 1O to get (the last equality in the display to follow is an exercise) Z ˜ −1 × h(1 − Ra)λ , 1Oi = 1O(x) − 1O(xa ) d x = ord(a). F × ˜ ˜ ˜ The calculation has shown that Raλ = λ − ord(a)δ. That is, {λ, δ} is the basis of a two-dimensional F ×-representation with the action given by λ˜ 1 −ord(a) λ˜ R = . a δ 0 1 δ Unlike ﬁnite-group representations, this representation is not semisimple: the sub- representation spanned by δ does not have a complement. The proof that dim(S0(χ)) = 1 for all χ in the nonarchimedean case is complete.

8. The Local Functional Equation 8.1. Additive characters. Consider a particular nontrivial unitary character ψ : F −→ T. That is, ψ(x + x0) = ψ(x)ψ(x0), x, x0 ∈ F, and the outputs of ψ are complex values of absolute value 1. We refer to such a character simply as additive. One choice of the particular character is ψ = exp ◦2πi Tr , F/Qp but this choice is not necessarily the optimal normalization; for our purposes its role is to show that there are any additive characters at all. From now on, let ψ denote some ﬁxed additive character of F , but it need not yet be normalized in any particular way. With ψ in place, deﬁne the Fourier transform on Schwartz functions, Z F : S −→ S, (Fϕ)(ξ) = ϕ(t)ψ(ξt) dt. F We understand the measure to be normalized so that it is self-dual, meaning that

FF = R−1, NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 17 i.e., (FFϕ)(x) = ϕ(−x) for all ϕ ∈ S and x ∈ F . The question of how to normalize the measure to produce the desired self-duality will be discussed soon. We remark for future reference that for any a ∈ F × (exercise),

FRa−1 = |a|RaF (equality of operators on S), so that consequently (exercise—the next display is not a trivial rewrite of the previous one) 0 RaF = |a|FRa−1 (equality of operators on S ). We invoke that any nontrivial additive character ψe of F is a dilation of the given one. That is, for some a ∈ F ×,

ψe = ψa where ψa = ψ ◦ a. We want to choose c ∈ R+ such that the scaled measure d˜∗ = c d∗ is self dual with respect to ψe, granting that d∗ is self-dual with respect to ψ. Compute ﬁrst that (exercise)

Fe1O = cRa1O, and then second that consequently (check the steps)

−1 2 −1 2 −1 FeFe1O = cFeRa1O = c|a| Ra−1 Fe1O = c |a| Ra−1 Ra1O = c |a| 1O, so that the right choice is c = |a|1/2. The conductor ν(ψ) of ψ is the negative of the exponent of the largest (as a set) P-power where ψ is trivial, ψ = 1 on P−ν(ψ) but not on P−(ν(ψ)+1). That is, the characterizing description of the conductor is ψ = 1 on P−e if and only if e ≤ ν(ψ). (The P-powers form a basis of open neighborhoods of 0 in F along with being subgroups.) It is an exercise to show that if the Schwartz function ϕ is a function on Pn/Pm then its Fourier transform Fϕ is a function on P−m−ν /P−n−ν where ν = ν(ψ). Given the conductor of ψ, we use the characterizing condition to determine the × conductor of ψa for any a ∈ F . For any e ∈ Z, −e −(e−ord(a)) ψa(P ) = ψ(P ), −e −(e−ord(a)) so that ψa = 1 on P if and only if ψ = 1 on P ), which in turn holds if and only if e − ord(a) ≤ ν(ψ), i.e., if and only if e ≤ ν(ψ) + ord(a). That is, by the characterizing condition,

ν(ψa) = ν(ψ) + ord(a). By choosing a suitably, we may replace our basic character by a dilation and then assume that the basic additive character has conductor 0, i.e., it is trivial on O but not on P−1. Granting that the basic additive character has conductor 0, the Fourier transform of the characteristic function of the integers is ( Z Z µ(O) if ψ is trivial on ξO, (F1O)(ξ) = 1O(t)ψξ(t) dt = ψξ(t) dt = F O 0 otherwise. 18 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

Since ψ has conductor 0, the calculation shows that F1O = µ(O)1O, and so we normalize the additive measure by specifying µ(O) = 1.

Now F1O = 1O, and consequently FF1O = 1O = R−1O. For any ψ, the Fourier inversion condition that FF = R−1 holds for exactly one normalization of the Haar measure. So in particular, when ψ has conductor 0, the relevant Haar measure is our normalization µ(O) = 1. And more generally, when ψ has conductor ν (so that o o ψ = ψ$ν where ψ is an additive character of conductor 0), the Haar measure is normalized by the condition µ(O) = |$ν |1/2 = q−ν/2.

8.2. The local functional equation. Lemma 3.6 in Kudla states that µ ∈ S0(χ) =⇒ Fµ ∈ S0(χ−1| · |). For the proof, recall that

FRa−1 = |a|RaF (equality of operators on S), 0 RaF = |a|FRa−1 (equality of operators on S ). The lemma follows from computing that for any a ∈ F × and ϕ ∈ S,

hRaFµ , ϕi = |a|hFRa−1 µ , ϕi = |a|hRa−1 µ , Fϕi = χ−1(a)|a|hµ , Fϕi = χ−1(a)|a|hFµ , ϕi.

0 s Recall that for any χ and s, the space S (χ | · | ) is spanned by µχ |·|s , where Z s × hµχ |·|s , ϕi = ((1 − R$−1 )ϕ)(x)χ(x)|x| d x, F × For any χ and s, replace χ by χ−1| · |1−s in the lemma to get that 0 s Fµχ−1|·|1−s ∈ S (χ | · | ) = Cµχ |·|s .

That is, since the Fourier transform is an isometry and hence doesn’t kill µχ−1|·|1−s , × Fµχ−1|·|1−s = ε(s, χ, ψ)µχ |·|s for some ε(s, χ, ψ) ∈ C .

8.3. Pre-normalizing the ε-factor calculation. For any s ∈ C, χ |·|s is ramified if and only if χ is ramified, because | · |s = 1 on O×. Consequently the basic o function ϕ from earlier, taken to 1 by µ, is the same for µχ |·|s as it is for µχ. Specifically, it is ( o 1O if χ is unramified, ϕ = −1 1O× · χ if χ is ramified. Apply the relation from the end of the previous section,

ε(s, χ, ψ)µχ |·|s = Fµχ−1|·|1−s , to the basic function, giving a formula for the local ε-factor, o (2) ε(s, χ, ψ) = hµχ−1|·|1−s , Fϕ i.

We will show that: To compute ε(s, χ, ψ) we may assume that χ is unitary. NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 19

To see this, apply (2) twice to get t o ε(s, χ | · | , ψ) = hµχ−1|·|−t |·|1−s , Fϕ i o = hµχ−1 |·|1−s−t , χ , Fϕ i = ε(s + t, χ, ψ). Thus we may assume that χ is unitary, as desired. Next we show that: To compute ε(s, χ, ψ) we may assume that ψ is normalized.

× Fix any a ∈ F , and let ψe = ψa. Recall that if the measure dx is self-dual with respect to ψ then |a|1/2dx is self dual with respect to ψe. Use (2) with ψe in place of ψ, and then at the end of the calculation use (2) again with the original ψ, o ε(s, χ, ψa) = hµχ−1|·|1−s , Feϕ i 1/2 o = |a| hµχ−1|·|1−s ,RaFϕ i 1/2 o = |a| hRa−1 µχ−1|·|1−s , Fϕ i 1/2 s−1 o = |a| |a| χ(a)hµχ−1|·|1−s , Fϕ i = |a|s−1/2χ(a)ε(s, χ, ψ). Thus we may assume that ψ is normalized, as desired. 8.4. The ε-factor calculation. From the previous section, we have the formula for ε(s, χ, ψ) o (3) ε(s, χ, ψ) = hµχ−1|·|1−s , Fϕ i and the reduction identity s−1/2 ε(s, χ, ψa) = |a| χ(a)ε(s, χ, ψ). If χ is unramiﬁed then ﬁrst take ψ normalized to have conductor 0. As discussed, we then have F1O = 1O, and so (3) becomes

ε(s, χ, ψ) = hµχ−1|·|1−s , 1Oi = 1. Combine the reduction identity with the previous display to obtain for any a ∈ F ×, s−1/2 ε(s, χ, ψa) = |a| χ(a).

The additive character ψa has conductor ν = ord(a), so rewrite the previous display, now letting ψ denote a general additive character of conductor ν,

ε(s, χ, ψ) = q(1/2−s)ν χ($ν ),F = na, χ unramiﬁed.

If χ is ramified then its conductor c = c(χ) ∈ Z+ is defined as the exponent of the largest (as a set) P-power such that χ is trivial on that P-power translate about 1, χ = 1 on 1 + Pc(χ) but not on 1 + Pc(χ)−1. (The P-power translates about 1 form a basis of open neighborhoods of 1 in F × along with being subgroups.) Let c ∈ Z+ denote the conductor of χ, and assume first that ψ has conductor 0. o −1 c The basic function ϕ = 1O× · χ is a function on O/P , and so its Fourier transform Fϕo is a function on P−c/O. In particular, (Fϕo)(ξ) = 0 if ord(ξ) < −c. 20 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

Next we show that also (Fϕo)(ξ) = 0 if ord(ξ) > −c. The Fourier transform is

Z o −1 (Fϕ )(ξ) = χ (t)ψξ(t) dt. O×

× −1 × If c = 1 then ord(ξ) ≥ 0 and so ψξ is trivial on O ; but χ is nontrivial on O , and so the previous display shows that (Fϕo)(ξ) = 0. (Since χ−1 is a multiplicative character but the integral involves additive Haar measure, we can not merely quote the result that the integral of a nontrivial character over a compact group vanishes until we realize that the additive Haar measure is also multiplicative Haar measure on the unit group.) If c > 1 then we have a natural isomorphism between a multiplicative group of additive cosets and a multiplicative group of multiplicative cosets,

(O/Pc−1)× −→∼ O×/(1 + Pc−1), u + Pc−1 7−→ u(1 + Oc−1).

Consequently

Z o X −1 (Fϕ )(ξ) = χ (u(1 + t))ψξ(u + t) dt c−1 u∈(O/Pc−1)× P Z X −1 −1 = χ (u)ψξ(u) χ (1 + t)ψξ(t) dt. c−1 u∈(O/Pc−1)× P

c−1 −1 c−1 But ψξ is trivial on P and χ is nontrivial on 1 + P , and so the integral is 0, again because we are integrating over a compact subgroup of the unit group, so that the additive Haar measure is also multiplicative. The remaining case is ξ = u$−c where u ∈ O×. The Fourier transform is

Z o −1 (Fϕ )(ξ) = χ (t)ψu$−c (t) dt O× Z −1 = χ(u) χ (ut)ψ$−c (ut) d(ut) O× Z = χ($cξ) χ−1(t)ψ($−ct) dt. O×

Rewrite the previous display as

(Fϕo)(ξ) = χ($cξ)q−c/2τ(χ, ψ), where τ(χ, ψ) is the normalized Gauss sum,

Z τ(χ, ψ) = qc/2 χ−1(t)ψ($−ct) dt. O× NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 21

(To see that the normalization is correct, compute that the square of the modulus of the integral is Z Z χ−1(t)χ−1(t0)ψ($−c(t − t0)) dt0 dt O× O× Z Z = χ−1(t)χ−1(tt0)ψ($−c(1 − t0)t) dt0 dt O× O× Z Z = χ−1(t0) ψ($−c(1 − t0)t) dt dt0 O× O× Z Z = χ−1(t0) ψ($−c(1 − t0)t) dt dt0 O× O Z Z − χ−1(t0) ψ($−c(1 − t0)t) dt dt0. O× P The first inner integral vanishes unless 1 − t0 ∈ Pc, in which case it is µ(O) = 1, and then the outer integral is µ(1 + Pc) = µ(Pc) = q−c. So overall the first double integral is q−c. The second double integral is similar except that now the inner integral vanishes unless 1 − t0 ∈ Pc−1, in which case the outer integral vanishes. Thus the modulus of the integral is q−c/2 as desired.) Now we have (Fϕo)(ξ) = q−c/2τ(χ, ψ)ϕo($cξ). And ϕo is the basic function for χ−1. Consequently the formula (3) for ε gives −c/2 o ε(s, χ, ψ) = q τ(χ, ψ)hµχ−1|·|1−s ,R$c ϕ i −c/2 o = q τ(χ, ψ)hR$−c µχ−1|·|1−s , ϕ i −c/2 −1 −c −c 1−s o = q τ(χ, ψ)χ ($ )|$ | hµχ−1|·|1−s , ϕ i = q−c/2τ(χ, ψ)χ($c)q(1−s)c = q(1/2−s)cτ(χ, ψ)χ($c). So far the calculation has assumed that ψ has conductor 0. The general case is o o ψ = ψ$ν where ψ has conductor 0. By the second reduction identity, ε(s, χ, ψ) = q(1/2−s)ν χ($ν )q(1/2−s)cτ(χ, ψo)χ($c) = q(1/2−s)(c+ν)χ($c+ν )τ(χ, ψo). To express the Gauss sum in terms of ψ rather than ψo, compute Z τ(χ, ψo) = qc/2 χ−1(t)ψo($−ct) dot O× Z 1/2(c+ν) −1 −c = q χ (t)ψ$−ν ($ t) dt O× Z = q1/2(c+ν) χ−1(t)ψ($−c−ν t) dt, O× and so the more general definition of the normalized Gauss sum is Z τ(χ, ψ) = q1/2(c+ν) χ−1(t)ψ($−c−ν t) dt. O× In sum,

ε(s, χ, ψ) = q(1/2−s)(c+ν)τ(χ, ψ)χ($c+ν ),F = na, χ ramiﬁed. 22 NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS

9. The Global ε-Factor Fix an archimedean place v. We have shown that the local ε-factor is

(1/2−s)(cv +νv ) cv +νv εv(s, χv, ψv) = qv τv(χv, ψv)χv($v ), where

qv = |O/P|,

$v = a generator of PFv,

cv = the conductor of χv,

νv = the conductor of ψv, ( 1 χv unramiﬁed, τv(χv, ψv) = 1/2(cv +νv ) 1/2(c+ν) R −1 −cv −νv qv q × χ (t)ψv($ t) dt χv ramiﬁed. Ov v v × Replace ψ by ψa where a ∈ k . At each nonarchimedean place we have from before, s−1/2 εv(s, χv, ψv,a) = |a|v χv(av)εv(s, χv, ψv). Assuming that the same formula holds for archimedean places, the global product of the local ε-factors is independent of ψ, and so it is written Y ε(s, χ) = εv(s, χv, ψv). v

Part 4. The Final Calculation To set up the ﬁnal calculation, deﬁne Y Λ(s, χ) = Lv(s, χv), v Y Zo(s, χ) = Zo,v(s, χv), v Y Z(s, χ) = Zv(s, χv). v Then

Λ(s, χ)Zo(s, χ) = Z(s, χ) multiplying together each Zv = (FZ)(1 − s, χ−1) by the global functional equation Y −1 = (FZv)(1 − s, χv ) decomposing FZ v Y −1 −1 = Lv(1 − s, χv )(FZo,v)(1 − s, χv ) factoring each FZv v Y −1 = Lv(1 − s, χv )εv(s, χv, ψv)Zo,v(s, χv) by the local functional equations v −1 = Λ(1 − s, χ )ε(s, χ)Zo(s, χ) assembling Λ, ε, and Zo. Therefore Λ(s, χ) = ε(s, χ)Λ(1 − s, χ−1). NUMBER FIELD ZETA INTEGRALS AND L-FUNCTIONS 23

Part 5. Application: Quadratic ε-Factors are Trivial Let k be a quadratic number ﬁeld, and let χ be its character. Note that χ−1 = χ. Compute that

Zk(1 − s) = Zk(s)

= ZQ(s)ΛQ(s, χ)

= ZQ(1 − s)ε(s, χ)ΛQ(1 − s, χ) = ε(s, χ)Zk(1 − s). Thus ε(s, χ) = 1. This result encompasses the value of the quadratic Gauss sum.